The D=6 string-string duality, crucial for allowing the interchanging of the roles of 4-D spacetime and string-world-sheet loop expansion, entails that there is an equivalence between the K-3 membrane action and the orbifold action. Here are some thoughts and reflections.
In the bosonic sector, the membrane action is:
Recall I derived the total action:
which is highly non-trivial since Clifford algebras are a quantization of exterior algebras. Applying to the Einstein-Minkowski fibre-bundle, we get via Gaussian matrix elimination, an expansion of via Green’s-functions, yielding the on-shell action of M-theory in the Witten gauge:
with the kappa symmetry term. With the metric on , and the corresponding coordinates with an antisymmetric 3-tensor. Hence, the worldvolume is:
The bosonic sector lives on the boundary of the open membrane: two copies of , which naturally couple to the U(1) connections .
Now, double dimensional reduction of the twisted supermembrane on: